Xavi's side was a 4-2 winner on aggregate over Napoli in the Round of 32 after dropping into the competition after being eliminated from the UEFA Champions League.
It will now meet the Turkish Super Lig giant, with the first leg at Camp Nou to be played on 11 March (AEDT).
Sevilla, winner of a record six trophies in this competition, will face David Moyes's West Ham United, while LaLiga rival Real Betis will play Eintracht Frankfurt from the Bundesliga.
Rangers, which shocked Borussia Dortmund in the previous round, takes on Red Star and RB Leipzig will play Spartak Moscow. Spartak's home leg must be played at a neutral venue because of the military operation in Ukraine ordered by Russia president Vladimir Putin.
The first legs will be played on 11 March (AEDT), with the exception of Porto versus Lyon and Betis' game with Frankfurt, which will be held a day earlier. The second legs will be played a week later.
UEFA Europa League Round of 16 draw
Rangers v Red Star
Braga v Monaco
Porto v Lyon
Atalanta v Bayer Leverkusen
Sevilla v West Ham United
Barcelona v Galatasaray
RB Leipzig v Spartak Moscow
Real Betis v Eintracht Frankfurt
